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I am reading the book The Evolution of Physics. I have a doubt in the topic "The field as representation". In this topic authors give the example of gravitational force represented as a field. In the following image the small circle represents an attracting body(say sun) and the lines are the well known lines of force of the gravitational field.
It is said that the density of the lines of force in space shows how the force varies with the distance. Let us consider a finite volume $\Delta V$ in the vicinity of sun. Now the number of lines of force passing through this is finite but there are infinite points in this $\Delta V$ volume.
$1.$Is there any gravitational force acting on those points through which no line of force passes.
$2.$ If the gravitational force acts on all the points contained in $\Delta V$ shouldn't there be infinite lines of forces passing through $\Delta V$.
$3.$ If it is supposed that there are really infinite lines of force passing thru $\Delta V$ then how to decide the density of no of lines won't it be infinite.
Please cite some canonical references which explains the 3 different points I've mentioned in your answer.